This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Foundations book using category theory?

I'm about to embark on a PhD in mathematical biology. My major is in computer science. I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research ...
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1answer
48 views

Universe enlargement and modal logic

In model theory and category theory, we often need to "enlarge" our universe (whatever that means) so that our proper classes become "small" and we can thereby manipulate them in more sophisticated ...
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0answers
47 views

Is it known if “Homotopy type theory” (HTT) can consistently model objects beyond V?

I read the free book on HTT but could not find an answer to this question. If that were the case HTT would be stronger than ZFC+LC (for any choice of the Large Cardinal axiom). That would be ...
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5answers
294 views

Can mathematics be traced back to a fundamental system of truths?

I'm not sure exactly how to state this question, or even if it belongs here. Still, I hope you will consider it, as I find it very interesting: Most of the results I've seen in mathematics come from ...
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2answers
105 views

Are arithmetic operators really sets in $\mathsf {ZFC}$?

Are two familiar arithmetic operators on $\mathbb N $, i.e. $+$ and $\cdot$, really functions in a set-theoretic sense? I.e., is '$1+2=3$' really '$\left ( \left ( 1, 2 \right ), 3 \right )\in+$'?
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1answer
54 views

Is there a way to phrase “there does not exist a universal set” in structural language?

Both ZFC (which is a good example of a material set theory) and ETCS (which is a good example of a structural set theory) prove the sentence "there is no set having maximum cardinality" as an easy ...
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1answer
24 views

Definition of an $n$-tuple agreeing with the Kuratowski's definition of an ordered pair

Is there a nice and elegant definition of an $n$-tuple ($n$ is a nonnegative integer) in ZFC, which will at the same time agree with the Kuratowski's definition of an ordered pair, i.e. $\left ( a,b ...
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1answer
37 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
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28 views

The role of verifiable computing in the formalization of mathematics

I've been thinking about this for a while, and it seems to me that mathematics "works" because (in principle) we can to check proofs very quickly, even though the discovery of that proof may have ...
2
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1answer
49 views

How do we decide which axioms are necessary?

I am studying the axioms for a complete ordered field. I have looked at different sources, some of which differ slightly in their listings. Given some construction of the reals (e.g. Dedekind cuts or ...
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1answer
126 views

What is a non-constructible real?

I am not sure to fully understand the concept (I read many of the wikipedia definitions for many of these issues but I am still confused). For instance, is a surreal number an example of ...
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2answers
106 views

Unbounded Finity?

Consider the successor of the largest finite ordinal that will ever be considered alone. But then it wasn't the largest finite ordinal that will ever be considered alone. How do we get around this ...
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2answers
87 views

Can $\mathbb R$ be enumerated after all?

I understand that the real numbers are innumerable, they cannot be put in bijection with the natural numbers. But assuming ZFC, a well-order of $\mathbb R$ can be achieved and thus there is a 'first' ...
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4answers
389 views

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
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1answer
102 views

When should I start learning Set Theory?

I started to learn a few disciplines on my own over the break after my first year in college and one of them was Real Analysis. In the process I came across many issues in Analysis texts concerning ...
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2answers
349 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
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0answers
46 views

Can a metatheory recognize objects from all permitted axiomatic systems?

Can there be/ has there been created a meta-theory that recognizes the existence of all objects whose existence is derivable from (or just "true in") each axiomatic system analyzable by the ...
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8answers
573 views

The line between axiom and theorem

Consider the intermediate value theorem. The theorem is very intuitive and may be described as obvious: if you go from $A$ to $B$ without teleporting, you have been everywhere between $A$ and $B$. ...
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1answer
21 views

Reformulation of Theories

Philosophical questions (or even just a matter of taste) regarding some mathematical constructions can give rise to reformulations of whole theories, for example, we can develop (Non-standard) ...
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1answer
46 views

What's the difference between the Well-ordering principle and Least Upper Bound property?

Well-Ordering Principle : Every non-empty set of positive integers contains a least element Least Upper Bound Property: Every nonempty bounded subset of the real numbers has a least upper bound. Is ...
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2answers
167 views

Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
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2answers
78 views

Do the ordinals exist before the universe of sets is constructed?

Should I worry about the following appeIarance of circularity in ZFC set theory? In constructing the universe of sets, you start with the empty set and then keep taking the power set over and over. ...
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3answers
122 views

Why isn't the inductive set _the_ set of natural numbers?

ZFC's axiom of infinity states: $$\exists x (\varnothing \in x \wedge \forall y \in x (y\cup \left \{y \right \} \in x)) $$ Isn't this set $ x $ really $\mathbb{N}$? It wouldn't be $\mathbb{N}$ if x ...
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0answers
25 views

Selecting a unique pair satisfying a condition $\varphi$ with an ordering

Given a finite structure $\mathfrak{A}$ with Universe $|A| < \infty$ and signature $\tau$. We say a pair $(a,a') \in A$ satisfies a $\tau$-formular $\varphi$ iff $$ \mathfrak{A} \models ...
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3answers
79 views

How does ZFC describe addition?

Surprisingly, the Wikipedia article on addition doesn't contain the answer. I looked elsewhere online for it, but didn't find it. Intuitively, the cardinal of the union of two sets seemed ...
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36 views

Does this conception of ordered pairs work? [duplicate]

Starting with the description of an ordered pair (a,b)=(c,d)<=>a=c, b=d. Could this be re-stated: Either, each element on the left is equal to an element on the right, which no other element on ...
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3answers
194 views

How does ZFC define functions?

I found the following definition on Wikipedia. Is it the most common definition? How is the definition usually notated? A function f from X to Y is a subset of the Cartesian product X × Y ...
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0answers
87 views

Unifying concepts in mathematics [closed]

Background Unfortunately my background's in engineering, so we've only been taught bits and pieces of math needed to be fluent in the science, but as I've started studying abstract algebra and real ...
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0answers
48 views

Proving triangles are unique, from Hilbert's axioms

I came across this problem in a book by Hartshorne on the foundations of geometry: Given a triangle ABC, show that the sides AB, AC and BC and the vertices A, B and C are uniquely determined by the ...
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1answer
118 views

What concepts does math take for granted?

I suspect there must be some concepts that math takes for granted (there has to be a starting point). For example, after spending some time thinking about it yesterday, I wondered whether most of ...
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1answer
79 views

Which mathematician has argued that we should move from “set” to “order”?

I recall reading (on this very site, in fact) that there is a mathematician whom argues that we ought to switch from "set" to "order" in the foundations, so as to "recover duality" or some such. Does ...
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1answer
50 views

Properties of Natural Numbers and Mathematical Induction

When working with natural numbers how to check that the property we consider is "permissible" to speak about? And not like the property "The smallest positive integer not definable in under eleven ...
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2answers
125 views

Doubt: Proof of existence of the summation of natural numbers in Landau

With reference to the Definition of Addition in Landau's Foundations of Analysis, the author, in proving the existence of a natural number $(x+y)$, takes for granted that $x' + y = (x+y)'$ where ...
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1answer
76 views

What is the intuition behind $\Delta_1^0$ sets and $\Delta_1^1$ sets?

In the context of first-order arithmetic, if $\phi$ is a formula with only bounded quantifiers, then if you put existential quantifiers in front it becomes a $\Sigma_1^0$ formula according to the ...
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1answer
38 views

What subsystem of second-order arithmetic can interpret the theory of real closed fields?

Real numbers can be encoded as sets of natural numbers, because they can be encoded as Dedekind cuts or Cauchy sequences of rational numbers, and a rational number can be encoded by a natural number. ...
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5answers
405 views

Ambiguity in the Natural Numbers

What I am wondering is if mathematicians know whether (assuming consistency) the natural numbers are a definite object, without ambiguity. This seems intuitively obvious, but I don't know if its been ...
5
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1answer
224 views

How best to formalize propositions suffering from “size issues”?

Suppose we want to formalize a proposition (say, from category theory or model theory) that has "size" issues. For concreteness, lets take the following statement as a fairly typical example. ...
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3answers
222 views

Why is zero important?

I am not sure whether this question is more appropriate here or in theoretical computer science. I leave it to the wisdom of moderators. On the computer science site I came across the following ...
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1answer
76 views

Has anyone successfully axiomatized the category of finite sets? In such a way that the resulting theory is bi-interpretable with PA.

In studies of ZFC, it is conventional to take Peano arithmetic (hereafter PA) as the metatheory. However, I don't like this convention; I think a better approach would be a metatheory (like ZF-fin ...
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1answer
81 views

How to describe free magmas in more structuralist terms?

Given a generating set $G$ (assume for simplicitly it consists entirely of urelements), the free magma on $G$ can be described concretely as follows. Its underlying set is the least $U \supseteq G$ ...
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1answer
82 views

Who first proved that the second-order theory of real numbers is categorical?

The second-order theory of real numbers is obtained by taking the axioms of ordered fields and adding a (Dedekind) completeness axiom, which states that every set which has an upper bound has a least ...
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47 views

What is the proof-theoretic strength of the predicative second-order theory of real numbers?

The first-order theory of real numbers, AKA the theory of real closed fields, is obtained by added to the axioms for ordered fields an axiom schema of completeness, which states that for each formula ...
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1answer
66 views

What is $M_x$ in Frege's Basic Law IIb?

Gottlob Frege's magnum opus, "The Basic Laws of Arithmetic" (Die Grundgesetze der Arithmetic in German) constitutes one of most impressive and meticulous attempts at developing a rigorous foundation ...
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1answer
78 views

Can equinumerosity by defined in monadic second-order logic?

Two properties (or concepts) $F$ and $G$ are said to be equinumerous if they have the same cardinality, i.e. if they can be put in one-to-one correspondence with each other. This can be very easily ...
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3answers
721 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
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0answers
137 views

Was Fermat's last theorem proved based on Peano's postulates?

Is the proof of Fermat's last theorem solely based on the Peano's postulates $+$ first order logic? Or it contains other axiomatic systems as well? What does it mean from foundations of math ...
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1answer
180 views

Can we capture all domains of discouse in the predicate logic within categorical logic?

In the construction of the bounded quantifiers via adjoints in the fibered category of subsets over a set (see e.g. here on Wikipedia), is there any restriction on the sets - specifically regarding ...
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1answer
308 views

What underlies formal logic (or math, generally)?

I read a useful definition of the word understanding. I can't recall it verbatim, but the notion was that understanding is 'data compression': understanding happens when we learn one thing that ...
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4answers
96 views

Free variables in definitions

Consider the epsilon delta definition of the limit below, as it is usually stated: the limit of f as f approaches a is L if and only if, for all ε > 0, there is a δ > 0 such that for all x, 0 ...
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1answer
58 views

What is the definition of the domain of composite partial functions?

In calculus books they define the domain(natural domain) of $f+g$ as $Dom(F)\bigcap Dom(g)$. And they define the domain of $fog$ as the set of all real numbers $x$ such that $x$ is in the domain of ...